Addition and Subtraction of real numbers (1.3 & 1.4)

Math 051 lecture notesProfessor Jason Samuels

ex) 3 + 5 =

ex) 42 + 29 =

ex) 12 - 4 =

ex) 7 - 9 =

ex) -3 - 4 =

ex) 6 - (-2) =

ex) -5 - (-3) =

ex) 7 + (-2) =

you do:ex) 12 - (-7) =

ex) -8 + 11 =

ex) -9 + 4 =

ex) 7 - 11 + 2 =

ex) [(7-5) - (11-6)] =

Multiplication and Division (1.6 & 1.7)

ex) 3 · 2 =

ex) (4)(-3) =

ex) (-3)(-6) =

you do:ex) (6)(-5) =

whats the operation?

operation: multiplicationsign - negative times negative is positivenegative times positive is negativepositive times negative is negative

Compare:

ex) 6 - 3 =ex) 6(-3) =

you do:ex) -7-4 =ex) (-7)(-4) =

some division:

ex) 12 ÷ 4 =ex) 21 ÷ (-7) =

ex) 20 = -4

ex) -16 = -8

Exponents

ex) (2)(2)(2)(2) can also be written as:ex) 36 is the same as: ex) 50 =

ex) 1 can also be written as:(6)(6)

order of operations(or, do them all together)

ex) 7 + 3 · 2 =

ex) (4)(-5) + (-3)(-6) =

ex) (3+6)4 - 8

you do:ex) 3 - 2(6 - 8) =

parentheses: might mean- do me first- multiply- sometimes both

opposite, reciprocal, absolute value (1.2)

adding what number doesnt change what you have?0 ("additive identity")

ex) what do you add to -3 to get 0 ?

thats called the additive inverse, or "opposite"

multiplying by what number doesnt change what you have?1 ("multiplicative identity")

ex) what do you multiply by 5 in order to get 1 ?1/5

thats called the multiplicative inverse, or "reciprocal"

ex) whats the opposite of 7/5 ?

whats the reciprocal?

ex) what is |3| ?

ex) |-3| =

ex) |2-5| =

Properties of real numbers (1.5)

commutative property

ex) 12+25 = 25+12

ex) (5)(7) = (7)(5)

associative property

ex) (9+5)+6 = 9+(5+6)

distributive property

ex) 3(4+11) = 3(4) + 3(11)

ex) -4(7-3) =

do it again... with fractions (1.9)Q. what is a fraction?-part of a whole-means of division-decimal-percent-rational number-one integer "over" another integer

there's a '3' in the '6' so thats all you need

why does adding fractions work like that?- here's the picture

why does fraction multiplication work like that?- here's the picture

why do we use variables?

- to represent (unknown) numbers- to help solve an equation- to show a general relationshipex) Sam is 18, Jason is 33

so, when Sam is 40, Jason iswhen Sam is x, Jason is

two very DIFFERENT situations: expressions & equations

we know expressions from arithmetic:ex) 4(7-5)what do you do? you calculate

note that your final result is a number - feels like an answerthere are expressions in algebra:ex) 3(2x-7) + 4what do you do? you simplify.[why?]

note that your final result has x in it - not very satisfyingheres an equation: x + 5 = 8

how are EXPRESSIONS and EQUATIONS different?

suppose you have 3x = 12what do you do?

you want to SOLVE FOR xyou get x=4you cant do that with expressions

ex) 5x+20what do you do?nothing (unless someone tells you to do something)

3x=12 ... what operation did we do?divide both sides by 3

can you do something like that with 5x+20 ?no: 5x+20 cannot magically become x+4if you have an expression, there is no "=", and it cant magically appear

Evaluating expressions

evaluate the following expressionsex) x2+4x if x=5

ex) 7 - x3 + 3x if x=-2

ex) 2(4x-5)+3 if x=4

ex) 1/2 gt2 if g=10, t=3

good general rule: use parentheses when you substitute (plug in) a value

ex) 4(x-3)2 if x=5

ex) x/4 + 4x if x=-8

a word about exponents, parentheses, and signs

ex) (-3)(-3)= ex) -3 · 3 =rewrite them with exponents:

(-3)2 -32

so these are NOT THE SAMEon the left, you square everything, including the negativeon the right, you only square the 3

ex) (-2)4 =

ex) (-x)4 =

lets talk about expressions (2.1)

ex) 2x - 6 + 4 + 3x

"pieces" are called termsthey are separated by addition and subtraction

compare:ex) 3 + x - 2 ex) 3x-2

what are "like terms" ?...terms you can combine

simplify:ex) 5+2x

ex) 2x+3x

ex) 3+2(4-3x)

ex) 4x+3[2(9-3x)+4]

now,ex) can you simplify x3 + 3x - 2 ?

are there any like terms?

ex) 5m2 + 3m - 2m2 + m

ex) 3x2 + 2x2

ex) 6t2 + 4 - 3t2 + t + 2

Solving linear equations (2.4, includes 2.2, 2.3)

what does it mean to be "linear" ?...if the expression only has x or y, nothing like x2 or xy2 (can be any variables, of course)

solve for the variable:ex) x + 2 = 7

ex) x + 24 = 47

ex) t - 8 = 27

ex) x- (-4) = 15

ex) 4 = x - 19

ex) 2x = 6

ex) 12x = 48

ex) x = 15 3

ex) 7x = 84

ex) x = 125 7

more solving for the variableex) 2x + 7 = 15

you want x by itself - what do you get rid of first?

ex) 3x - 5 = 5x + 9

ex) 4(x+2) = 2(3x+1)

you do:ex) -4x-3 = 13 ex) x+7 = 7x+31 ex) 2(4-x)+4 = 3(2x-4)

now, with fractionsex) 2 x + 3 = 11

3

ex) 18 x = 720100

you do:ex) 11 x = 55

20

parentheses are in the way- so get rid of them ... how?- by distributing

make it so that x appears on only one side of the equation

solving for a variable, when there are two variables (2.5)

if you are given the value for one variableex) 2x - 5y = 30 solve for y if x=3

(thats just like the other problems)ex) 3x + 2y = 12 solve for x

actually, this works exactly the sameto get x by itself, what do you need to get rid of?

ex) h + 4 = 5 solve for h 3

ex) x = 6 solve for xy

ex) s = 4 solve for tt

ex) y+3 = 4 solve for y x

problem: t is in the denominatorfix it: multiply by t on both sides

hw questions

2.2#19

Solving word problems (2.6, 2.7)

you have to TRANSLATE between words and math

ex) you have three. then you get two more. how many do you have now?

ex) Dave has a certain amount. he gets seven more. how many does he have now?

ex) Alan has a certain amount. it triples. how much does he have now?

ex) Tanya has a certain amount. it doubles. then she gets four more. how much does she have now?

ex) How much do Dave, Alan and Tanya have together?

percents

ex) 50% of 12 is what?

ex) 15% of 80 is what?

ex) 20% of what is 15?

ex) at a restaurant, you got great service and want to give a 20% tip on a $60 bill. how much is the tip?

ex) at the pharmacy you buy some aspirin. the sales tax on the purchase was $2. sales tax rate is 8%. how much was the aspirin sticker price?

you can do the calculations using decimals or fractionsby hand, i think fractions are easierby calculator, decimals are easier.....its your choice

ex) a rectangle has a length 5 inches longer than it is wide. the perimeter is 58 inches. what are the dimensions of the rectangle?

ex) Joel has twice as many nickels as dimes, all together worth a total of $1.80 how many does he have of each?

first, give each unknown a namenext, write down the relationship

ex) two consecutive integers add to 45. what are the numbers?

inequalities (2.8, 2.9)

ex) solve for x: 2x = 6

graph the solution on the number line:

now,ex) solve for x: 2x > 6

solve it exactly the same way

graph the solution on the number line:

ex) 3x-7 < 1

everything is the same, with ONE EXCEPTIONwhich is bigger?ex) 3__4multiply both sides by (-2)ex) -6__-8

so, when you multiply (or divide) by a negative number, the inequality switchesnote that this is the only timeex) 3<4, subtract 1 from both sides, you still have 2<3

ex) -3x < 15

compound inequalities [optional]

how do we write "x is between 2 and 5" ?thats the same as saying that its bigger than 2, and also less than 5we could write: x>2 and x<5

heres a shorter way:take this 2 < xcombine it with this x < 5to get this 2 < x < 5

solve:ex) 2 < 3x-7 < 14

its easy, do them both at the same time

hw questions

2.6#17 jack is twice as old as lacey. in three years the sum of their ages will be 54. how old are they now?

2.6#37 tanner has $4.35 in nickels and quarters if he has 15 more nickels than quarters, how many of each does he have?

linear equations, representing and graphing (3.1-5)

ex) cell phone monthly bill

Shyanna's bill: $39.99 (and all the calls she wants)Tarik's bill: 10cents/minJerry's bill: $29.99 plus 5cents/min

which plan is best?...depends how much you talk

what about 250min?Shyanna: $39.99 note: in terms of dollars "10 cents" is ".10"Tarik: (250)(.10) = $25Jerry: (.05)(250) + 29.99 = 12.50 + 29.99 = $42.49

what about 550 min?Shyanna: $39.99Tarik: (.10)(550) = $55Jerry: (.05)(550) + 29.99 = 27.50 + 29.99 = $57.49

how can we represent this information?

- numerically# minutes Shyanna Tarik Jerry0 39.99 0 29.99250 39.99 25 42.99350 39.99 35 47.99450 39.99 45 52.99550 39.99 55 57.99

- graphically

Shyanna Tarik Jerry

how much does it cost to talk 0 minutes for:Shyanna? Tarik? Jerry?

what does that have to do with the graph?...where x=0 is where the graph crosses the y-axis - that is called the y-intercept

Tarik pays .10 dollars per min, and he also pays $10 for 100 minutesis that the same thing?how can we check?

in fact, we can do this with any two points and this tells us the rate, or slopeex)

ex)

note that this is different from calculating y/x ...thats assumes that we started at (0,0), which may not be true[economics sometimes uses this calculation]

for Shyanna's plan, what is her rate? is it always the same?

for Tarik's plan, what is his rate? is it always the same?

for Jerry's plan, what is his rate? is it always the same?

a linear function has the same rate (or slope) everywhere

what is the function for each plan?Shyanna: Tarik:

Jerry:

in general, for a linear function, we can writey = mx + b

m = slopeb = y-intercept

examples of linear functions

ex) y = 3x+2graph: slope=

y-intercept=

some solutions: x | y

ex) y = 2x-1

ex) y = -2x+3

if you have two points, call them (x1 y1) and (x2 y2)

slope = change in y = y2 - y1

change in x x2 - x1 also called rise/run

if slope is negative,@ line is decreasing (going down)@ the graph will go "down & right" (not down & left)

solutions can be written as table or ordered pair

also equations of a line:ex) 2x + 3y = 12graph: slope:

y-intercept:solutions:

ex) x - 3y = 6graph: slope:

y-intercept:solutions:

finding the equation of a line (3.6)

find the equation of a line given the slope and y-interceptex) slope=3, y-intercept=4

(pretty easy)

find the equation of a line given the slope and any pointex) slope=2, point is (3,4)method 1: use y=mx+bwe know the slope, so we have y = 2x+bwe can plug in the values from the point and it will satisfy the equation

4 = 2(3) + b4 = 6 + b-2 = b

so the equation is: y = 2x - 2

method 2:the answer is y - 4 = 2(x-3)first, where does that come from?second. how can there be two different answers?

second question first: its really the same formula:y-4 = 2(x-3)y-4 = 2x-6y = 2x-2 ...aha!

now, where did that come from?...it comes from the slope (here's the explanation, if you are curious)we know that the slope is 2

2 = y2 - y1

x2 - x1 but we have values for one point, so (x1 y1) = (3,4)2 = y2 - 4 x2 - 3also, we want this to be true for any point on the line, that is (x,y)2 = y - 4 x - 3cross-multiply to gety - 4 = 2(x-3)

or, just remember the formula:y - y1 = m(x - x1)

ex) find the equation of the line with slope=-4, through the point (5,2)

find the equation of the line using two pointsex) find the equation of the line through (2,7) and (4,13)we can find the slope:

now we have the slope and a point (pick either one), so its like the problem we just solved

you do:ex) find the equation of the line through (-2,3) and (1,9)

special cases

-horizontal linesgraph:

whats the slope?

whats the equation?

-vertical linesgraph:

whats the slope?

whats the equation?

note that the slope is undefined - you cannot write the equation of this line as y=mx+b

hw questions ch3

Solving TWO linear equations with TWO variablesthree ways: by graphing (4.1), by elimination (4.2), by substitution (4.3)

here's what a problem might look like:ex) suppose you sell concessions at a cinema. you sell small sodas for $2 and large for $3. you ran out of small cups, so you only have one size cup today (for a small order, you fill it halfway). at the end of the day, you sold 86 sodas for $191. your manager wants to know how many of each size you sold. can you tell him?

what do we know?

the situation:- two variables- two equations...what do you do?

simplify the situation so you have one variable and one equation (...once you have that, you know what to do)

was that magic? no, lets learn how to do it.

ex) x = 6 solve for x & yy = 2x+3

(thats too easy)ex) y = x-2

y = 2x-6

solve with algebra:

solve with a graph:

the solution tells us the values for x,y that work in both equations

ex) y = 4x-3 solve for x and yy = x+9

ex) 2x+y = 7 solve for x and yy = 4x-11 how do we solve this?

ex) x - 2y = -73x + 2y = 3

do it again, a quicker wayex) x - 2y = -7

3x + 2y = 3

thats too easy - are you allowed to do that?consider: a = 3

b = 2a+b = ?

c - d = 7 d = 2 c = ?

you do:ex) 2x - 3y = 5

x + 3y = 7

note: when you solve for the second variable, you need to plug a value into an equation. you have two choices, it doesnt matter which equation you use

now,ex) 2x - y = 9

x + 2y = 2 what do you need to have so that one variable gets eliminated?

ex) 2x + 5y = 94x + 5y = 9

you do:ex) 2x + 3y = 5

3x + y = 11

heres the key step:to eliminate x, we want the 'x' in the second equation to have a coefficient of '-2'so, multiply by -2whatever you do to one side of the equation, you do to the other, so its legal

note: some tricky person might put "y" first.......make sure x's are lined up and y's are lined up

when is it easier to you use....

substitution? ...when one variable is already by itself

elimination? ...when one variable is set up to cancel(the coefficients are same number, opposite sign)

ex) 2x - 4y = -10 solve for x & y3x + 2y = 1

special cases:

ex) x+2y=3x+2y=4

ex) 2x-3y=44x-6y=8

~~~same thing, with word problems (4.4)

ex) John has $1.70 in dimes and nickels with 22 total coins. how many of each does he have?

ex) at a cinema, adult tickets are $12, child tickets are $7. if 110 tickets are sold for $1150, how many adults and children came?

ex) the difference between two numbers is 12. their sum is 38. what are the numbers?

ex) one number is one more than triple another number. their sum is 29. what are the numbers?

ex) two consecutive integers add to 45. what are the numbers?

ex) two consecutive even integers add to 74. what are the numbers?

hw questions

4.1#1x+y=3 solve by graphingx-y=1

when there are ( ), raise everything in the ( ) to the exponentwhen there are no ( ), only raise the number (or variable) without the sign

combining for addition: same base, same exponentcombining for multiplication: same base

not -9

once you 'flip it'you have taken care of the negative

"8 subtract 5" ... 8 - 5"8 subtract -5" ... 8 + 5 if you "stack it up" and you are doing

subtraction, remember to flip the signs

ex) start with 3x4+4x2-7x+4 and subtract -4x4+6x3+3x2-4x+2

ex) simplify: (x2 - 3x + 2) - (3x2 + 5x - 3)

ex) subtract x2-4x+7 from 3x2-2x-6

note that "subtract A from B" means B - Awatch out!

5.6 special cases

ex) (x+4)2

ex) (s-5)2

ex) (3x+2)2

ex) (4x - 7y)2

5.7 dividing a polynomial by one term

ex) 3x4 + 2x3 - 7x2

x2

ex) 4x5 - 10x3 + 8x2x

you do:ex) 12x5 + 9x4 + 15x3

3x2

ex) 10x4 + 20x3 - 15x2

5x2

ex) 16x4 - 20x3 + 16x2

4x

hw questions ch5

5.7#72

review for midterm

ch6 - Factoring

opposite of multipliying out

recall:ex) multiply 3x(x+4)

answer: 3x2 + 12xex) multiply (x+5)(x-7)

answer: x2 - 2x - 35now:ex) given 3x2 + 12x, get 3x(x+4) ex) given x2 - 2x - 35, get (x+5)(x-7)

how do we do that?

6.1 factoring - find the gcf

factor:ex) x2 + 3x

ex) 3x4 + 7x3 - 4x2

ex) 4p2 + 12p + 20

ex) 6r3 - 18r2 + 6r

ex) 6a2b3 + 2ab4 + 2a4b2

look at:coefficiencts"a" exponents"b" exponents

you do: factorex) 8x3 - 16x5 + 20x2

ex) 12r3s4 - 6rs5

6.2 some basic factoring

x2 + 5x + 6 = ( ?? )·( ?? )how do we do this?first we make an observation: the lead term is x2

... (what) times (what) will give x2 x2 + 5x + 6 = (x___)(x___)

how do you do this?...first, look at last coefficient, figure what two numbers multiply to that (many possiblities) next, look at middle coefficient, find which pair add to that

ex) x2 + 6x + 8

ex) x2 - x - 12

ex) x2 + 2x - 8

ex) x2 - 49

ex) x2 + 10x + 25

note: sometimes you CANNOT factorex) x2 + 2x + 8try to factor it:

ex) x2 + 4x + 7

...and when i say "CANNOT", i mean you dont know how. you can factor this with advanced techniques which are beyond the scope of this course

HOW TO FACTOR (so far)

1. find gcf (greatest common factor)

2. x2 → "x and x"

3. product: break last term down into factor pairs

4. sum: look for the pair which adds/subtracts to the middle term

5. check

ex) 6r3 - 18r2 + 12r

you do ... factor:ex) 2x2 + 18x + 40

ex) 4x2 + 4x - 24

6.3 factoring - when the lead coefficient is NOT 1...a little harder

ex) 6x2 + 19x + 10

1. always check for a gcf first [here there is none]

2. multiply first and last coefficient

3. factor that into number pairs, look for which ones add to the middle number

4. rewrite the original polynomial by breaking up the middle term

5. for the first two terms and the last two terms, separately factor the gcf

6. factor again (if you did it right, the same factor will appear twice)

lets check the answer:

ex) factor 4x2 + 16x + 15

how will question be asked on an exam?

on the final: "factor completely"

on the Compass: "which of the following is a factor of 4x2 + 16x + 15 ?

you doex) factor 10x2 - x - 3

dont forget the +1check your work by multiplying

factoring - with two variables

how do you factor 6x2 + xy - 2y2 ...do it as though there is one variable

ex) factor 4x6 + 12x3y + 5y2

recall:ex) factor x2 - 81now,ex) factor 4x2 - 25

ex) factor 9r2 - 16

ex) factor 64x2 - 9y2

hw questions

6.2-30) factor 2a2 - 18a + 28

using factoring to solve equations

ex) solve for x: (x-2)(x+3) = 0what happens if you try to get x by itself?

we need a new method...solve for x: (x-2)(x+3) = 0

something times something equals zero...so one of them must be zero - aha!x-2=0 or x+3=0x=2 or x=-3

you get two solutions!

how can we use that in other sorts of problems?ex) solve for x: x2 + x - 6 = 0

moral of the story:suppose you have an equation where you must solve for xif the equation is NOT LINEAR (like if there is x2), you cannot get x by itselfthe way to solve this is to FACTOR

solve for x:ex) x2 - x - 12 = 0

ex) 2x2 - x - 10 = 0

ex) x2 + 2x - 7 = 1

ex) x3 - 5x2 + 4x = 0

a review of factoring:

ex) factor x2 + 2x - 24

ex) factor 6x2 + 11x - 10

solve for the variable:ex) s2 + 2s = -1

ex) 8x3 - 18x2 - 18x = 0

notice that this problem:2x + 3 = 5xis different from:2x2 + 3 = 5x...to do the second one you need to factor(that little 2 makes a big difference)

hw questions - ch6

ch7 rational expressions

7.1 reducing

we call this "canceling" but actually we are dividing

terms: things we are addingfactors: things we are multiplying

cant "cancel" x...why?cant cancel 2 and 6 ... why?

factoring gives you something times something, so you can divide and cancel

reduce:

reduce:

reduce:

see what happens when you multiply by (-1)ORfactor out (-1) directly

if there are *signs* and you are not sure, multiply one factor by (-1) and see if it gives you the other factor

COMPASS TIP: check your answer by plugging in a number

check by plugging in a number:

7.2 rational expressions: multiply and divide

first step: combine fractions...do you need LCD?second step: simplify

Same denominator - just add the numerators

7.4 rational expressions: solve equations

hw questions ch7

homework questions ch7

ch8 roots and radicals

what is a radical?...the opposite of a power

roots and signs

combine and simplify radicals

8.5 multiply, divide8.3 simplify8.4 add, subtract

for multiplication, you do not need like terms to combinefor addition, you dosame idea here, its just that the terms are more complicated (because of the radicals)

hw questions ch8

7.6 complex fractions

how do we simplify this?...well, "fraction" is the same as "division", so rewrite it as division - then solve it as you normally would

7.7 proportions and rates

ex) if a car travels 120 miles in 2 hours, how far would it go in 5 hours (at the same rate)?

ex) if Sam can read 240 pages in 3 nights, how much can she read in 8 nights (at the same rate)?

ex) on a map, 1 inch represents 85 miles. 3.5 inches represents how many miles?

ex) a baseball player gets 15 hits in 20 games. at that rate, how many hits will he have in 84 games?

after how many games will he have 78 hits?

distance

what is the distance between two points?...to find it we use the pythagorean formulaex) what is the distance between (2,4) and (6,9) ?

c2 = a2 + b2

to make this a formula for just the distance, we write:c = √a2 - b2

what is a?

what is b?

so, the distance is c =

lets make that one formula:for two points (x1 y1) and (x2 y2), the distance between them is c = √(x2 - x1)2+(y2 - y1)2

ex) find the distance between (5,2) and (4,6)

ex) find the distance between (4, -1) and (-3,2)

check your signs!

hw questions 7.6, 7.7

review for final